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252                      ANALYTICAL MECHANICS
r, . .       ~ .          energy expended in driving
Driving efficiency = ~~~----------~J~T~^'
to                        total energy expended
      , .        .           energy expended in def ormine;
Smashing efficiency =-----,& , T ^---------------T~I-
total energy expended
Consider the case of a blow which drives a nail or a pile. Let M be the mass of the hammer, m the mass of the pile, v the velocity of the hammer just before impact, V the velocity just after impact. The contact between the hammer and the pile may be regarded as inelastic, therefore just after the impact both the pile and the hammer have the same velocity v'. In other words, immediately after the impact there is an amount of energy equal to | (M + m) z/2 available for driving the pile, while the balance of the energy of the hammer, that is, | Mv2  \ (M + m) z/2, is expended during the impact in producing permanent deformation, heat, and sound. Substituting these in the two definitions for the efficiency of a blow we obtain
TV . .       ~ .            (M + m) v'2 Driving efficiency = -r~-----
Q      , .       ~ .           1     (M + m)?/2
bmasnine; eniciency = 1 --------r-----
Immediately after the impact practically all the momentum of the hammer relative to the earth will be in the hammer and the pile; therefore we can write
Mv= (M+m) vf.
Eliminating the velocities between the last equation and the above expressions for the two efficiencies we obtain
Driving efficiency =
Smashing efficiency =
M+m                 (xii)
It is evident from these expressions that for driving piles or nails the ram or the hammer head must have a large mass